Local and nonlocal Laplacian diffusion on graphs with obstacles

We study the effect of obstacles on diffusion processes on finite path graphs from a linear algebraic perspective. An obstacle is modeled as a weighted self-loop on the path graph, and diffusion is governed by local and nonlocal Laplacian operators. We first analyze the spectral perturbation induced by the obstacle for the standard graph Laplacian and prove that, as the weight of the self-loop increases, the diffusive particle becomes effectively confined to a subregion of the graph. Surprisingly, we show that the same trapping phenomenon occurs for a broad class of nonlocal operators given by matrix functions, including fractional powers of the Laplacian. Although these operators are nonlocal, they inherit structural properties that prevent them from overcoming arbitrarily strong obstacles. In contrast, we prove that transformed d-path Laplacians—particularly the Mellin-transformed operator—overcome obstacles of arbitrary weight. This distinction is explained through spectral decomposition and block diagonalization techniques. Our results establish a sharp structural difference between matrix functions and transformed d-path Laplacians and provide a general framework for modeling nonlocal diffusion on graphs with barriers.